Ising Chain with Several Phase Transitions

Abstract

The one‐dimensional spin‐½ Ising model with very long‐range ferromagnetic interaction and first and second neighbor antiferromagnetic interactions is solved exactly in a direct magnetic field. A study of the finite temperature behavior confirms the existence of up to four first‐order phase transitions ending in classical critical points. We find a confluence of three critical lines at a ``tricritical'' point with exponents $\mathrm{\beta =}\frac{1}{4}{\text{, δ=5, γ=γ}}^{\prime }\text{=1,α=0}$ , and ${\alpha }^{\prime }=\frac{1}{2}$ . We find also a confluence of four critical lines at a ``tetracritical'' point with exponents $\mathrm{\beta =}\frac{1}{4}{\text{, δ=7, γ=γ}}^{\prime }\text{=1, α=0}$ , and ${\alpha }^{\prime }=\frac{2}{3}$ . We show the existence of triple lines and ``quadruple'' lines, along which three or four phases are simultaneously in contact with each other, respectively. For some special values of the interaction strengths, five phases are in contact with each other at a ``quintuple'' point. We remark that our ``multiple'' lines and ``multicritical'' points are not ``true'' multiple lines and multicritical points but that they would be so in a somewhat extended model. This model satisfies a generalized form of the ordinary phase rule of Gibbs. In distinction to various models considered recently, our model does not exhibit a singular phase‐boundary diameter.